Infinitary logic, large cardinals and AECs: some reflections Andrs - - PowerPoint PPT Presentation

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Infinitary logic, large cardinals and AECs: some reflections

Andrés Villaveces - Universidad Nacional de Colombia - Bogotá Reflections on Set Theoretical Reflection Bagaria 60 - Montseny - Catalonia - November 2018

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Contents

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Part 1 A Catalan Prelude

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

A Lullian Prelude - Logic in Catalonia in the 13th Century Logic in Catalonia has an illustrious history.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

A Lullian Prelude - Logic in Catalonia in the 13th Century Logic in Catalonia has an illustrious history. Reflection Principles have been in the mind of Catalan Logicians for a long time, in different forms, at different times.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

A Lullian Prelude - Logic in Catalonia in the 13th Century Logic in Catalonia has an illustrious history. Reflection Principles have been in the mind of Catalan Logicians for a long time, in different forms, at different times. In the 13th Century: Llull.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Llull: Logic in Catalonia in the 13th Century

from Atlas Català

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Llull: Logic in Catalonia in the 13th Century

(detail)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Llull: epistemology and ontology of the universe

◮ Synthesis of the three cultures thriving (Sefarad, Al-Andalus, Hispania) ◮ Descriptions of the world through strong images: ◮ Trees (of Science), Spheres (of Predicates) and

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Llull: epistemology and ontology of the universe

◮ Synthesis of the three cultures thriving (Sefarad, Al-Andalus, Hispania) ◮ Descriptions of the world through strong images: ◮ Trees (of Science), Spheres (of Predicates) and ◮ Reflection of Imago Dei through Imago Mundi, ◮ The universe as a system of categories reflecting one another but ◮ anchored in different “models” reflecting one another. (Superposition of planes, all of them reflecting the original, “divine” plane...)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Ramon Llull - A Scale

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Reflections / Above reflecting below / Similitude

But let us listen directly to Llull:

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Reflections / Above reflecting below / Similitude

But let us listen directly to Llull: ◮ Entre semblança i semblança ha disposició e fi e proporció e concordança... (car) totes semblances en cors sustentades són de una comuna semblança.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Reflections / Above reflecting below / Similitude

But let us listen directly to Llull: ◮ Entre semblança i semblança ha disposició e fi e proporció e concordança... (car) totes semblances en cors sustentades són de una comuna semblança. ◮ ... ciències esteses en moltes veritats, ço és saber, en lurs semblances (...); el seu encercament està en pujant o en davallant de les coses dejús a les dessús e de les dessús a les dejús, e en los efectus d’aquelles han d’elles coneixença...

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

New logics? - Abulafia / Llull

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Llull

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Part 2 Reflection Classes (Beyond Syntax...)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Origins: infinitary logic

One of the questions that started the process was the problem of proving Categoricity Transfer, a Morley-like theorem, for the infinitary logic Lω1,ω.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Origins: infinitary logic

One of the questions that started the process was the problem of proving Categoricity Transfer, a Morley-like theorem, for the infinitary logic Lω1,ω. Namely, is it true that if an Lω1,ω-sentence ψ is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals?

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Origins: infinitary logic

One of the questions that started the process was the problem of proving Categoricity Transfer, a Morley-like theorem, for the infinitary logic Lω1,ω. Namely, is it true that if an Lω1,ω-sentence ψ is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals? More generally, what is the behavior of the function I(ψ, λ) := |{M | = ψ | |M| = λ}/ ≈ |, for a sentence ψ of the logic Lω1,ω?

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Long story short

After many attempts, the analysis of that primal question ran off from the syntactic extreme (infinitary logic(s)) to a more semantic “extreme”.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Long story short

After many attempts, the analysis of that primal question ran off from the syntactic extreme (infinitary logic(s)) to a more semantic “extreme”. The attempts: ◮ (Keisler) Use “sequentially homogeneous” models. But sequential homogeneity is a consequence of categoricity... ◮ (Shelah) The role of models of size ℵn (n < ω) in the decomposition of large models, the role of dimension-like

• bstructions.

◮ (Shelah) Forcing-like approach to types that would eventually become “Galois types”.

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“Algebraically-minded model theory” - Really?

Another early origin of Abstract Elementary Classes, complementary to the Categoricity problem, was Shelah’s idea of (as expressed in his paper The Lazy Model-Theoretician’s Guide to Stability Theory 1973)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

“Algebraically-minded model theory” - Really?

Another early origin of Abstract Elementary Classes, complementary to the Categoricity problem, was Shelah’s idea of (as expressed in his paper The Lazy Model-Theoretician’s Guide to Stability Theory 1973) speaking mainly to “those who are interested in algebraically-minded model theory, i.e., generic models, the class of e-closed models and universal-homogeneous models rather than elementary classes and saturated models. These were his words in 1975. He continues: “our main point is that though stability theory was developed for the latter context, almost everything goes through in the wider context (with suitable changes in the definitions).”

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What goes through, really?

This declaration (the “almost everything goes through”) entailed more than it could seem at first sight: in many ways it is true but it took a long time to build up the right notions of stability, of types, of independence.

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Smooth Reflection Classes

Replacing formulas by an abstract notion of “strong embedding” between L-structures is the first important point. In the definition of AECs we do not declare membership in the class by satisfying some sentence or some axiomatic system.

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Smooth Reflection Classes

Replacing formulas by an abstract notion of “strong embedding” between L-structures is the first important point. In the definition of AECs we do not declare membership in the class by satisfying some sentence or some axiomatic system. The relation | =, basic in First Order logic, takes a back seat here, and the main relation ≤K (a generalization of the elementary submodel relation ≺ of first order) now leads the game.

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Formalism Freeness?

All of this approach very much goes in line with other situations in mathematics where versions of “Formalism Freeness” (Kennedy) take up center stage. One of them is computability (Turing, Post, Gödel, Kleene, Church). Another

• ne is Model Theory as a “generalized Galois theory”, as

happens in AECs. Kennedy, Magidor, Väänänen: Inner Models for Different

• Logics. Yet another take on formalism freeness, dual to

Abstract Elementary Classes.

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The definition [Abstract Elementary Class]

Fix a language L. A class K of L-structures, together with a binary relation ≤K on K is an abstract elementary class (for short, AEC) if: 1. Both K and ≤K are closed under isomorphism. This means two things: first, if M′ ≈ M ∈ K then M′ ∈ K; second, if M′, N′ are L-structures with M′ ⊂ N′, M′ ≈ M, N′ ≈ N and M ≤K N then M′ ≤K N′. 2. If M, N ∈ K, M ≤K N then M ⊂ N, 3. ≤K is a partial order, 4. (Coherence) If M ⊂ N ≤K N′ and M ≤K N′ then M ≤K N, 5. (LS) There is a cardinal (called “the Löwenheim-Skolem number” of the class) κ = LS(K) ≥ ℵ0 such that if M ∈ K and A ⊂ |M|, then there is N ≤K M with A ⊂ |N| and |N| ≤ |A| + LS(K), 6. (Unions of ≤K -chains) If (Mi)i<δ is a ≤K -increasing chain of length δ (δ a limit ordinal), then

◮

i<δ(Mi)i<δ ∈ K,

◮ for each j < δ, Mj ≤K

• i<δ Mi,

◮ if for each i < δ, Mi ≤K N ∈ K then

i<δ Mi ≤K N.

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Main Conjecture: The Main Gap The Main Gap Theorem for FO logic

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Main Conjecture: The Main Gap The Main Gap Theorem for FO logic

The “gold standard” of mathematical logic, of model theory, in various ways, and the main conjecture in AECs.

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At this point, we have the following situation: ◮ So far, no control on possible axiomatization of the class K. The emphasis is placed on its being closed under the constructions specified in the axioms. However, later (in subsection) we focus

• n the logical control of these classes. Remember Shelah’s

“algebraically-minded model theory”. ◮ These are not necessarily amalgamation classes: there is no amalgamation axiom. However, many AECs do satisfy the amalgamation property. Furthermore, the model theory will depend on the kind of amalgamation possible in the class.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Part 3 Back to syntax!

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

How to deal with these AECs?

Theorem (Presentation Theorem, Shelah)

Let (K, ≤K) be an AEC in a language L. Then there exist ◮ A language L′ ⊃ L, with size LS(K), ◮ A (first order) theory T ′ in L′ and ◮ A set of T ′-types, Γ′, such that K = PC(L, T ′, Γ′) := {M′ ↾ L | M′ | = T ′, M′ omits Γ′}. Moreover, if M′, N ′ | = T ′, they both omit Γ′, M = M′ ↾ L and N = N ′ ↾ L, M′ ⊂ N ′ ⇔ M ≤K N.

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Corollary (“Hanf” number of an AEC)

If an AEC K has a model of cardinality ≥ (2LS(K))+ then it has arbitrarily large models.

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Corollary (“Hanf” number of an AEC)

If an AEC K has a model of cardinality ≥ (2LS(K))+ then it has arbitrarily large models. Proof: Use the Hanf number for PC classes (and the undefinability of well-ordering).

• Theorem (Shelah)

Let (K, ≤K) be an AEC with amalgamation and arbitrarily large

• models. If K is categorical in λ > LS(K) then it is µ-galois-stable for

each cardinal µ ∈ [LS(K), λ).

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Corollary (“Hanf” number of an AEC)

If an AEC K has a model of cardinality ≥ (2LS(K))+ then it has arbitrarily large models. Proof: Use the Hanf number for PC classes (and the undefinability of well-ordering).

• Theorem (Shelah)

Let (K, ≤K) be an AEC with amalgamation and arbitrarily large

• models. If K is categorical in λ > LS(K) then it is µ-galois-stable for

each cardinal µ ∈ [LS(K), λ). And many other results on Stability Theory for a.e.c.’s really hinge

• n this “syntactic embedding” of EM models coming from First

Order logic! The proof hinges on Ehrenfeucht-Mostowski models (whose existence in AECs with large enough models is given by the Presentation Theorem).

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Beyond the Presentation Theorem

The complexity of Projective Classes in the Presentation Theorem (the enlarged language) may be avoided at the price of axiomatizing an AEC via a sentence in a different logic.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Beyond the Presentation Theorem

The complexity of Projective Classes in the Presentation Theorem (the enlarged language) may be avoided at the price of axiomatizing an AEC via a sentence in a different logic. First response: Shelah (and Boney-Vasey): under categoricity of a proper class of cardinals, and a.e.c. K may be axiomatized by a sentence in L∞,ω.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Beyond the Presentation Theorem

The complexity of Projective Classes in the Presentation Theorem (the enlarged language) may be avoided at the price of axiomatizing an AEC via a sentence in a different logic. First response: Shelah (and Boney-Vasey): under categoricity of a proper class of cardinals, and a.e.c. K may be axiomatized by a sentence in L∞,ω. More generally...

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The logic L1

κ

A logic called L1

κ (Shelah, 2007), in between Lκ,ω and Lκ,κ (for κ

singular strong limit): Lκ,ω ⊂ L1

κ ⊂ Lκ,κ

that has many desirable properties:

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

The logic L1

κ

A logic called L1

κ (Shelah, 2007), in between Lκ,ω and Lκ,κ (for κ

singular strong limit): Lκ,ω ⊂ L1

κ ⊂ Lκ,κ

that has many desirable properties: ◮ Undefinability of well-order (very weak compactness) ◮ Interpolation (“balancing” the interpolation problem between Lκ,ω and Lκ,κ): if φ → ψ ∈ Lκ,ω, φ has vocabulary L1, ψ has vocabulary L2 then there is θ in the common vocabulary L1 ∩ L2 such that φ ⊢ θ ⊢ ψ... BUT θ ∈ Lκ,κ. ◮ Downward Löwenheim-Skolem ◮ Maximality for the previous properties (“Lindström”): any logic above Lκ,ω satisfying undefinability of well-order, occurrence below κ (for κ = κ strong limit) interpolation and LS must be ≤ L1

κ.

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The connection with a.e.c.’s

(Work in progress, with Shelah) For any a.e.c. K with τ = τK, κ = LSTK, λ = 2(κ + |τ|)+ there exists ψK ∈ Lλ+,κ+(τ) such that K = Mod(ψK).

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

The connection with a.e.c.’s

(Work in progress, with Shelah) For any a.e.c. K with τ = τK, κ = LSTK, λ = 2(κ + |τ|)+ there exists ψK ∈ Lλ+,κ+(τ) such that K = Mod(ψK). ψK is in the same vocabulary as the class!!! (This provides some interesting return, some interesting symmetry to Kennedy’s description of a.e.c.’s in terms of Formalism Freeness!)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

The connection with a.e.c.’s

(Work in progress, with Shelah) For any a.e.c. K with τ = τK, κ = LSTK, λ = 2(κ + |τ|)+ there exists ψK ∈ Lλ+,κ+(τ) such that K = Mod(ψK). ψK is in the same vocabulary as the class!!! (This provides some interesting return, some interesting symmetry to Kennedy’s description of a.e.c.’s in terms of Formalism Freeness!) Moreover, ψK ∈ L1

κ∗,

≺K≈≺L1

κ∗

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Part 4 Back to syntax, really?

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Really, back to syntax??

There are many issues related to L1

κ:

◮ No actual definition of the syntax

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Really, back to syntax??

There are many issues related to L1

κ:

◮ No actual definition of the syntax ◮ No “Consistency Properties” attached to the logic

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Really, back to syntax??

There are many issues related to L1

κ:

◮ No actual definition of the syntax ◮ No “Consistency Properties” attached to the logic ◮ Only partial understanding of its expressive power

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

On the “Delayed” Ehrenfeucht-Fraïssé game

The syntax is really defined in terms of an Ehrenfeucht-Fraïssé “partial equivalence” game

G

Γ,θ,α(M1, M2):

◮ Player I chooses a sequence from M1,

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

On the “Delayed” Ehrenfeucht-Fraïssé game

The syntax is really defined in terms of an Ehrenfeucht-Fraïssé “partial equivalence” game

G

Γ,θ,α(M1, M2):

◮ Player I chooses a sequence from M1, ◮ Player II breaks the sequence into ω parts and chooses a sequence in M2,

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

On the “Delayed” Ehrenfeucht-Fraïssé game

The syntax is really defined in terms of an Ehrenfeucht-Fraïssé “partial equivalence” game

G

Γ,θ,α(M1, M2):

◮ Player I chooses a sequence from M1, ◮ Player II breaks the sequence into ω parts and chooses a sequence in M2, ◮ Player I acts following the challenge from the breakup, on the FIRST piece and plays another sequence,

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

On the “Delayed” Ehrenfeucht-Fraïssé game

The syntax is really defined in terms of an Ehrenfeucht-Fraïssé “partial equivalence” game

G

Γ,θ,α(M1, M2):

◮ Player I chooses a sequence from M1, ◮ Player II breaks the sequence into ω parts and chooses a sequence in M2, ◮ Player I acts following the challenge from the breakup, on the FIRST piece and plays another sequence, ◮ Player II acts following the challenge from the breakup, on the SECOND piece and plays another sequence,

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Infinite debts, finite time

This game has been described in terms

• f “Infinite debts,

finite time to pay them off”.

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

Infinite debts, finite time

This game has been described in terms of “Infinite debts, finite time to pay them off”. The point: Playing the game, I “opens up” space for possible answers - possible functions - and “simulates” the role of the expansion by predicates from the Presentation Theorem. Mysteries: ◮ a strong syntax for L1

κ,

◮ info from a.e.c.’s?

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Examples of sentences in L1

λ+

◮ Let cf λ > ω, φn(x1, ..., xn) ∈ L1

λ for all n < ω. Then

∃(xα)α<λ

• α1,...αn<λ

φn(xα1, ..., xαn) is in L1

λ+.

◮ So, the sentence ∃(xα)α<ω1

• α<β<ω1(xα < xβ) is in L1

ω2.

◮ (Väänänen) Under GCH, there is a sentence of L1

ℵω2 which has a

model of cardinality ℵα for α < ℵ2 if and only if cf (α) = ω1.

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A “Lindström theorem” for L1

κ

Using (Strong) Undefinability of Well Ordering

Theorem (Shelah’s “Lindström theorem” for L1

κ)

Suppose κ = κ. Then L1

κ is the unique logic L such that:

◮ L ⊂ H(κ) ◮ L has the < κ-occurrence property ◮ Lκω ≤ L ◮ L has strong undefinability of well-order

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A “Lindström theorem” for L1

κ

Using (Strong) Undefinability of Well Ordering

Theorem (Shelah’s “Lindström theorem” for L1

κ)

Suppose κ = κ. Then L1

κ is the unique logic L such that:

◮ L ⊂ H(κ) ◮ L has the < κ-occurrence property ◮ Lκω ≤ L ◮ L has strong undefinability of well-order A corollary of this result is Interpolation: if κ = κ, φ1, a τ1-sentence and φ2, a τ2-sentence (both of L1

κ) are such that |

= φ1 → φ2 then there is a (τ1 ∩ τ2)-sentence ψ of L1

κ such that |

= φ1 → ψ → φ2.

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The logics L1

κλ+.

(Joint work with Väänänen) Suppose cf (λ) > ω and 2λ < κ. Given Φ a function (I → φI) from P(λ) to formulas such that φI has free variables among xα, α ∈ I, let ∀ x

• f
• n

φf −1(n)( x′) and ∃ x

• f
• n

φf −1(n)( x′) with x a sequence of length λ, f : λ → ω, n ∈ ran(f ), x′ the subsequence of x corresponding to indices in f −1(n).

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A descending chain of length λ

The sentence ∃ x

• f
• n
• {xα < xβ | f (α) = f (β) = n, β < α}

says the linear order < has a descending chain of length λ

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A modified delayed EF game, DGδ(M, N)

Similar to the original delayed EF game except the partitioning by player II is finer: First round of the game:

I II δ0 < δ {a0

α : α < λ}

f0 : λ → ω n0 {b0

α : f0(α) = n0}

(1)

Position after first round: (δ0, {(a0

α, b0 α) : f0(α) = n0}).

If δ0 = 0, the game ends and II wins if h0 = {(a0

α, b0 α) : f0(α) = n0}

is a partial isomorphism. Otherwise I wins.

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Second round of DGδ(M, N):

I II δ1 < δ0 {a1

α : α < λ}

f1 : λ → ω n1 {b1

α : f1(α) = n1}

(2)

Again, if δ1 = 0, the game ends and II wins if h0 ∪ h1, where h1 = {(a1

α, b1 α) : f1(α) = n1},

is a partial isomorphism. Otherwise I wins. The position now is (δ1, h0 ∪ h1). The game continues until δi = 0.

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Syntax

Proposition

The relation “II has a winning strategy in DGδ(M, N)" is transitive. The syntax may be reconstructed from formulas of the form ψδ

N, a

that capture the notion M ≡δ

L1

∞λ+ N

(originally defined as II having a winning strategy for the game DGδ(M, N)).

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In conclusion...

◮ Smooth Reflection Classes (or Abstract Elementary Classes: emphasis on pure reflection phenomena as opposed to definability)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

In conclusion...

◮ Smooth Reflection Classes (or Abstract Elementary Classes: emphasis on pure reflection phenomena as opposed to definability) ◮ Yet logic makes a (cruel but sweet) return, first through the Presentation Theorem (at the price of having to use the Σ1

1

• peration).

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

In conclusion...

◮ Smooth Reflection Classes (or Abstract Elementary Classes: emphasis on pure reflection phenomena as opposed to definability) ◮ Yet logic makes a (cruel but sweet) return, first through the Presentation Theorem (at the price of having to use the Σ1

1

• peration).

◮ For different reasons (Lindström-motivated) there was a very nice logic, strictly in between Lκω and Lκκ: L1

κ when κ is a

singular limit

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

In conclusion...

◮ Smooth Reflection Classes (or Abstract Elementary Classes: emphasis on pure reflection phenomena as opposed to definability) ◮ Yet logic makes a (cruel but sweet) return, first through the Presentation Theorem (at the price of having to use the Σ1

1

• peration).

◮ For different reasons (Lindström-motivated) there was a very nice logic, strictly in between Lκω and Lκκ: L1

κ when κ is a

singular limit ◮ This logic, however, still lacks syntax in the traditional sense (but we are working on fixing that)

A Catalan Prelude Reflection Classes (Beyond Syntax...) Back to syntax! Back to syntax, really?

In conclusion...

◮ Smooth Reflection Classes (or Abstract Elementary Classes: emphasis on pure reflection phenomena as opposed to definability) ◮ Yet logic makes a (cruel but sweet) return, first through the Presentation Theorem (at the price of having to use the Σ1

1

• peration).

◮ For different reasons (Lindström-motivated) there was a very nice logic, strictly in between Lκω and Lκκ: L1

κ when κ is a

singular limit ◮ This logic, however, still lacks syntax in the traditional sense (but we are working on fixing that) ◮ A variant of that logic now has syntax; another variant captures (without the Σ1

1 operation) the phenomenon of Smooth

Reflection

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Moltes gràcies! Feliç aniversari, Joan!